Axiom:Axiom of Pairing/Set Theory/Strong Form

Axiom

For any two sets, there exists a set to which only those two sets are elements:

$\forall a: \forall b: \exists c: \forall z: \paren {z = a \lor z = b \iff z \in c}$

That is, let $a$ and $b$ be sets.

Then there exists a set $c$ such that $c = \set {a, b}$.

Thus it is possible to create a set whose elements are two sets that have already been created.

Also known as

The axiom of pairing is also known as the axiom of the unordered pair.

Some sources call it the pairing axiom.

Also see

• Equivalence of Formulations of Axiom of Pairing

Sources

• 1964: Steven A. Gaal: Point Set Topology ... (previous) ... (next): Introduction to Set Theory: $1$. Elementary Operations on Sets
• 1982: Alan G. Hamilton: Numbers, Sets and Axioms ... (previous) ... (next): $\S 4$: Set Theory: $4.2$ The Zermelo-Fraenkel axioms: $\text {ZF3}$

• Weisstein, Eric W. "Zermelo-Fraenkel Axioms." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Zermelo-FraenkelAxioms.html
• Weisstein, Eric W. "Axiom of the Unordered Pair." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/AxiomoftheUnorderedPair.html